- Open Access
Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays
© Xu et al.; licensee Springer. 2014
- Received: 12 February 2014
- Accepted: 7 July 2014
- Published: 4 August 2014
In this paper, we deal with a discrete Lotka-Volterra predator-prey model with time-varying delays. For the general non-autonomous case, sufficient conditions which ensure the permanence and global stability of the system are obtained by using differential inequality theory. For the periodic case, sufficient conditions which guarantee the existence of a unique globally stable positive periodic solution are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.
MSC:34K20, 34C25, 92D25.
- Lotka-Volterra predator-prey model
- global attractivity
After the pioneering work of Berryman  in 1992, the dynamic relationship between predators and their preys has become one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Dynamic nature (including the local and global stability of the equilibrium, the persistence, permanence and extinction of species, the existence of periodic solutions and positive almost periodic solutions, bifurcation and chaos and so on) of predator-prey models has been investigated in a number of notable studies [2–26]. In many applications, the nature of permanence is of great interest. For example, Fan and Li  made a theoretical discussion on the permanence of a delayed ratio-dependent predator-prey model with Holling-type functional response. Chen  addressed the permanence of a discrete n-species delayed food-chain system. Zhao and Jiang  focused on the permanence and extinction for a non-autonomous Lotka-Volterra system. Chen  analyzed the permanence and global attractivity of a Lotka-Volterra competition system with feedback control. Zhao and Teng et al.  established the permanence criteria for delayed discrete non-autonomous-species Kolmogorov systems. For more research on the permanence behavior of predator-prey models, one can see [32–44].
where denotes the density of prey species at time t, and stand for the density of predator species at time t, and (). Using Krasnoselskii’s fixed point theorem and constructing the Lyapunov function, Lv et al. obtained a set of easily verifiable sufficient conditions which guarantee the permanence and global attractivity of system (1.1).
Many authors have argued that discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have non-overlapping generations. Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulations [4, 16, 46]. Thus it is reasonable and interesting to investigate discrete time systems governed by difference equations. The principal object of this article is to propose a discrete analogue system (1.1) and explore its dynamics.
where for , .
which is a discrete time analogue of system (1.1), where .
For the point of view of biology, we shall consider (1.4) together with the initial conditions (). The principal object of this article is to explore the dynamics of system (1.4) applying the differential inequality theory to study the permanence of system (1.4). Using the method of Lyapunov function, we investigate the global asymptotic stability of system (1.4).
We assume that the coefficients of system (1.4) satisfy the following:
(H1) , , with are non-negative sequences bounded above and below by positive constants.
The remainder of the paper is organized as follows. In Section 2, basic definitions and lemmas are given, some sufficient conditions for the permanence of system (1.4) are established. In Section 3, a series of sufficient conditions for the global stability of system (1.4) are included. The existence and stability of system (1.4) are analyzed in Section 4. In Section 5, we give an example which shows the feasibility of the main results. Conclusions are presented in Section 6.
where is a non-negative sequence bounded above and below by positive constants. In order to obtain the main result of this paper, we shall first state the definition of permanence and several lemmas which will be useful in the proof of the main result.
Definition 2.1 
Lemma 2.1 
Lemma 2.2 
Now we state our permanence result for system (1.4).
In view of (2.4), (2.10), (2.15), (2.20), (2.25) and (2.30), we can conclude that system (1.4) is permanent. The proof of Theorem 2.1 is complete. □
Remark 2.1 Under the assumption of Theorem 2.1, the set is an invariant set of system (1.4).
In this section, we formulate the stability property of positive solutions of system (1.4) when all the time delays are zero.
for all .
Thus (3.4) holds true and the proof is complete. □
In this section, we further assume that () and the coefficients of system (1.4) satisfy the following condition:
(H5) There exists a positive integer ω such that for , , ().
Theorem 4.1 Assume that (H1)-(H5) are satisfied, then system (1.4) with all the delays () admits at least one positive ω-periodic solution which we denote by .
Clearly, F depends continuously on . Thus F is continuous and maps the compact set into itself. Therefore, F has a fixed point. It is not difficult to see that the solution passing through this fixed point is an ω-periodic solution of system (1.4). The proof of Theorem 4.1 is complete. □
Theorem 4.2 Assume that (H1)-(H5) are satisfied, then system (1.4) with all the delays () has a globally stable positive ω-periodic solution.
Proof Under assumptions (H1)-(H5), it follows from Theorem 4.1 that system (1.4) with all the delays () admits at least one positive ω-periodic solution. In addition, Theorem 3.1 ensures that the positive solution is globally stable. Hence the proof. □
In this paper, we have investigated the dynamic behavior of a discrete Lotka-Volterra predator-prey model with time-varying delays. Sufficient conditions which ensure the permanence of the system are established. Moreover, we also analyze the global stability of the system with all the delays () and deal with the existence and stability of the system. We have shown that delay has important influence on the permanence of the system. Therefore, delay is an important factor to decide the permanence of the system. When all the delays are zero, we obtain some sufficient conditions which guarantee the global stability of the system. Computer simulations are carried out to explain our main theoretical results.
The first author was supported by the National Natural Science Foundation of China (No. 11261010), the Soft Science and Technology Program of Guizhou Province (No. 2011LKC2030), the Natural Science and Technology Foundation of Guizhou Province (J2100), the Governor Foundation of Guizhou Province (53) and the Doctoral Foundation of Guizhou University of Finance and Economics (2010). The second author was supported by the National Natural Science Foundation of China (No. 11101126). The third author was supported by the Natural Science Innovation Team Project of Guizhou Province (14). The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper.
- Berryman AA: The origins and evolution of predator-prey theory. Ecology 1992, 73(5):1530-1535. 10.2307/1940005View ArticleGoogle Scholar
- Dai BX, Zou JZ: Periodic solutions of a discrete-time diffusive system governed by backward difference equations. Adv. Differ. Equ. 2005., 2005: Article ID 586218Google Scholar
- Gyllenberg M, Yan P, Wang Y: Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems. Physica D 2006, 221(2):135-145. 10.1016/j.physd.2006.07.016MathSciNetView ArticleGoogle Scholar
- Fan M, Wang K: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Model. 2002, 35(9-10):951-961. 10.1016/S0895-7177(02)00062-6View ArticleGoogle Scholar
- Fazly M, Hesaaraki M: Periodic solutions for a discrete time predator-prey system with monotone functional responses. C. R. Math. Acad. Sci. Paris 2007, 345(4):199-202. 10.1016/j.crma.2007.06.021MathSciNetView ArticleGoogle Scholar
- Gaines RE, Mawhin JL: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin; 1997.Google Scholar
- Kar TK, Ghorai A: Dynamic behaviour of a delayed predator-prey model with harvesting. Appl. Math. Comput. 2011, 217(22):9085-9104. 10.1016/j.amc.2011.03.126MathSciNetView ArticleGoogle Scholar
- Sen M, Banerjee M, Morozov A: Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect. Ecol. Complex. 2012, 11: 12-27.View ArticleGoogle Scholar
- Haque M, Venturino E: An ecoepidemiological model with disease in predator: the ratio-dependent case. Math. Methods Appl. Sci. 2007, 30(14):1791-1809. 10.1002/mma.869MathSciNetView ArticleGoogle Scholar
- Braza PA: Predator-prey dynamics with square root functional responses. Nonlinear Anal., Real World Appl. 2012, 13(4):1837-1843. 10.1016/j.nonrwa.2011.12.014MathSciNetView ArticleGoogle Scholar
- Wiener J: Differential equations with piecewise constant delays. Lecture Notes in Pure and Applied Mathematics 90. In Trends in Theory and Practice of Nonlinear Differential Equations. Dekker, New York; 1984.Google Scholar
- Xu R, Chen LS, Hao FL: Periodic solution of a discrete time Lotka-Volterra type food-chain model with delays. Appl. Math. Comput. 2005, 171(1):91-103. 10.1016/j.amc.2005.01.027MathSciNetView ArticleGoogle Scholar
- Zhang JB, Fang H: Multiple periodic solutions for a discrete time model of plankton allelopathy. Adv. Differ. Equ. 2006., 2006: Article ID 90479Google Scholar
- Xiong XS, Zhang ZQ: Periodic solutions of a discrete two-species competitive model with stage structure. Math. Comput. Model. 2008, 48(3-4):333-343. 10.1016/j.mcm.2007.10.004MathSciNetView ArticleGoogle Scholar
- Zhang RY, Wang ZC, Chen YM, Wu JH: Periodic solutions of a single species discrete population model with periodic harvest/stock. Comput. Math. Appl. 2009, 39(1-2):77-90.MathSciNetView ArticleGoogle Scholar
- Zhang WP, Zhu DM, Bi P: Multiple periodic positive solutions of a delayed discrete predator-prey system with type IV functional responses. Appl. Math. Lett. 2007, 20(10):1031-1038. 10.1016/j.aml.2006.11.005MathSciNetView ArticleGoogle Scholar
- Zhang ZQ, Luo JB: Multiple periodic solutions of a delayed predator-prey system with stage structure for the predator. Nonlinear Anal., Real World Appl. 2010, 11(5):4109-4120. 10.1016/j.nonrwa.2010.03.015MathSciNetView ArticleGoogle Scholar
- Li YK, Zhao KH, Ye Y: Multiple positive periodic solutions of species delay competition systems with harvesting terms. Nonlinear Anal., Real World Appl. 2011, 12(2):1013-1022. 10.1016/j.nonrwa.2010.08.024MathSciNetView ArticleGoogle Scholar
- Sun YG, Saker SH: Positive periodic solutions of discrete three-level food-chain model of Holling type II. Appl. Math. Comput. 2006, 180(1):353-365. 10.1016/j.amc.2005.12.015MathSciNetView ArticleGoogle Scholar
- Ding XH, Liu C: Existence of positive periodic solution for ratio-dependent N -species difference system. Appl. Math. Model. 2009, 33(6):2748-2756. 10.1016/j.apm.2008.08.008MathSciNetView ArticleGoogle Scholar
- Chakraborty K, Chakraborty M, Kar TK: Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay. Nonlinear Anal. Hybrid Syst. 2011, 5(4):613-625. 10.1016/j.nahs.2011.05.004MathSciNetView ArticleGoogle Scholar
- Li ZC, Zhao QL, Ling D: Chaos in a discrete population model. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 482459Google Scholar
- Xiang H, Yang KM, Wang BY: Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks. Discrete Dyn. Nat. Soc. 2005, 2005(3):281-297. 10.1155/DDNS.2005.281View ArticleGoogle Scholar
- Gopalsamy K: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht; 1992.View ArticleGoogle Scholar
- Kuang Y: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York; 1993.Google Scholar
- Fan L, Shi ZK, Tang SY: Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters. Nonlinear Anal., Real World Appl. 2010, 11(1):341-355. 10.1016/j.nonrwa.2008.11.016MathSciNetView ArticleGoogle Scholar
- Fan YH, Li WT: Permanence for a delayed discrete ratio-dependent predator-prey model with Holling type functional response. J. Math. Anal. Appl. 2004, 299(2):357-374. 10.1016/j.jmaa.2004.02.061MathSciNetView ArticleGoogle Scholar
- Chen FD: Permanence of a discrete n -species food-chain system with time delays. Appl. Math. Comput. 2007, 185(1):719-726. 10.1016/j.amc.2006.07.079MathSciNetView ArticleGoogle Scholar
- Zhao JD, Jiang JF: Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system. J. Math. Anal. Appl. 2004, 299(2):663-675. 10.1016/j.jmaa.2004.06.019MathSciNetView ArticleGoogle Scholar
- Chen FD: The permanence and global attractivity of Lotka-Volterra competition system with feedback control. Nonlinear Anal., Real World Appl. 2006, 7(1):133-143. 10.1016/j.nonrwa.2005.01.006MathSciNetView ArticleGoogle Scholar
- Teng ZD, Zhang Y, Gao SJ: Permanence criteria for general delayed discrete nonautonomous n -species Kolmogorov systems and its applications. Comput. Math. Appl. 2010, 59(2):812-828. 10.1016/j.camwa.2009.10.011MathSciNetView ArticleGoogle Scholar
- Dhar J, Jatav KS: Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories. Ecol. Complex. 2013, 16: 59-67.View ArticleGoogle Scholar
- Liu SQ, Chen LS: Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with distributed delay. Appl. Math. Lett. 2003, 16(6):911-917. 10.1016/S0893-9659(03)90016-4MathSciNetView ArticleGoogle Scholar
- Liao XY, Zhou SF, Chen YM: Permanence and global stability in a discrete n -species competition system with feedback controls. Nonlinear Anal., Real World Appl. 2008, 9(4):1661-1671. 10.1016/j.nonrwa.2007.05.001MathSciNetView ArticleGoogle Scholar
- Hu HX, Teng ZD, Jiang HJ: On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls. Nonlinear Anal., Real World Appl. 2009, 10(3):1803-1815. 10.1016/j.nonrwa.2008.02.017MathSciNetView ArticleGoogle Scholar
- Muroya Y: Permanence and global stability in a Lotka-Volterra predator-prey system with delays. Appl. Math. Lett. 2003, 16(8):1245-1250. 10.1016/S0893-9659(03)90124-8MathSciNetView ArticleGoogle Scholar
- Kuniya T, Nakata Y: Permanence and extinction for a nonautonomous SEIRS epidemic model. Appl. Math. Comput. 2012, 218(18):9321-9331. 10.1016/j.amc.2012.03.011MathSciNetView ArticleGoogle Scholar
- Hou ZY: On permanence of Lotka-Volterra systems with delays and variable intrinsic growth rates. Nonlinear Anal., Real World Appl. 2013, 14(2):960-975. 10.1016/j.nonrwa.2012.08.010MathSciNetView ArticleGoogle Scholar
- Li CH, Tsai CC, Yang SY: Analysis of the permanence of an SIR epidemic model with logistic process and distributed time delay. Commun. Nonlinear Sci. Numer. Simul. 2012, 17(9):3696-3707. 10.1016/j.cnsns.2012.01.018MathSciNetView ArticleGoogle Scholar
- Chen FD, You MS: Permanence for an integrodifferential model of mutualism. Appl. Math. Comput. 2007, 186(1):30-34. 10.1016/j.amc.2006.07.085MathSciNetView ArticleGoogle Scholar
- Berezansky L, Baštinec J, Diblík J, Šmarda Z: On a delay population model with quadratic nonlinearity. Adv. Differ. Equ. 2012., 2012: Article ID 230 10.1186/1687-1847-2012-230Google Scholar
- Baštinec J, Berezansky L, Diblík J, Šmarda Z: On a delay population model with a quadratic nonlinearity without positive steady state. Appl. Math. Comput. 2014, 227: 622-629.MathSciNetView ArticleGoogle Scholar
- Tang XH, Cao DM, Zou XF: Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay. J. Differ. Equ. 2006, 228(2):580-610. 10.1016/j.jde.2006.06.007MathSciNetView ArticleGoogle Scholar
- Tang XH, Zou XF: Global attractivity in a predator-prey system with pure delays. Proc. Edinb. Math. Soc. 2008, 51: 495-508.MathSciNetView ArticleGoogle Scholar
- Lv X, Lu SP, Yan P: Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments. Nonlinear Anal., Real World Appl. 2010, 11(5-6):574-583.MathSciNetView ArticleGoogle Scholar
- Chen YM, Zhou ZF: Stable periodic of a discrete periodic Lotka-Volterra competition system. J. Math. Anal. Appl. 2003, 277(1):358-366. 10.1016/S0022-247X(02)00611-XMathSciNetView ArticleGoogle Scholar
- Chen FD: Permanence for the discrete mutualism model with time delays. Math. Comput. Model. 2008, 47(3-4):431-435. 10.1016/j.mcm.2007.02.023View ArticleGoogle Scholar
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